We study the dimer-dimer scattering length $a_4$ for a two-component Fermi mixture in which the different fermions have different masses $mus$ and $mds$. This is made in the framework of the exact field theoretical method. In the large mass ratio domain the equations are simplified enough to lead to an analytical solution. In particular we link $a_4$ to the fermion-dimer scattering length $a_3$ for the same fermions, and obtain the very simple relation $a_4=a_3/2$. The result $a_4 simeq a_3/2$ is actually valid whatever the mass ratio with quite good precision. As a result we find an analytical expression providing $a_4$ with a fairly good precision for any masses. To dominant orders for large mass ratio it agrees with the literature. We show that, in this large mass ratio domain, the dominant processes are the repeated dimer-dimer Born scatterings, considered earlier by Pieri and Strinati. We conclude that their approximation, of retaining only these processes, is a fairly good one whatever the mass ratio.
We consider the problem of obtaining the scattering length for a fermion colliding with a dimer, formed from a fermion identical to the incident one and another different fermion. This is done in the universal regime where the range of interactions is short enough so that the scattering length $a$ for non identical fermions is the only relevant quantity. This is the generalization to fermions with different masses of the problem solved long ago by Skorniakov and Ter-Martirosian for particles with equal masses. We solve this problem analytically in the two limiting cases where the mass of the solitary fermion is very large or very small compared to the mass of the two other identical fermions. This is done both for the value of the scattering length and for the function entering the Skorniakov-Ter-Martirosian integral equation, for which simple explicit expressions are obtained.
